The (two) core of a hypergraph is the maximal collection of hyperedges within which no vertex appears only once. It is of importance in tasks such as efficiently solving a large linear system over GF, or iterative decoding of low-density parity-check codes used over the binary erasure channel. Similar structures emerge in a variety of NP-hard combinatorial optimization and decision problems, from vertex cover to satisfiability.
For a uniformly chosen random hypergraph of m=nρ vertices and n hyperedges, each consisting of the same fixed number l≥3 of vertices, the size of the core exhibits for large n a first-order phase transition, changing from o(n) for ρ>ρc to a positive fraction of n for ρ<ρc, with a transition window size Θ(n−1/2) around ρc>0. Analyzing the corresponding “leaf removal” algorithm, we determine the associated finite-size scaling behavior. In particular, if ρ is inside the scaling window (more precisely, ρ=ρc+rn−1/2), the probability of having a core of size Θ(n) has a limit strictly between 0 and 1, and a leading correction of order Θ(n−1/6). The correction admits a sharp characterization in terms of the distribution of a Brownian motion with quadratic shift, from which it inherits the scaling with n. This behavior is expected to be universal for a wide collection of combinatorial problems.
"Finite size scaling for the core of large random hypergraphs." Ann. Appl. Probab. 18 (5) 1993 - 2040, October 2008. https://doi.org/10.1214/07-AAP514